Some Problems on the Dynamics of Positive Characteristic Tori

Simple item page
dc.contributor.advisorBauer, Mark
dc.contributor.advisorNguyen, Dang Khoa
dc.contributor.authorGunn, Keira
dc.contributor.committeememberGreenberg, Matthew
dc.contributor.committeememberScheidler, Renate
dc.date2024-11
dc.date.accessioned2024-05-16T20:51:30Z
dc.date.available2024-05-16T20:51:30Z
dc.date.issued2024-05-13
dc.description.abstractThe positive characteristic tori T_F are a set of counterparts to the real torus T=R/Z. In positive characteristic we define the ``integers'' as polynomials with coefficients from a finite field F (Z_F:=F[t]) and the ``reals'' as the field of Laurent series with coefficients in F (R_F:=F((t))) so that the positive characteristic torus over F is similarly defined: T_F:=R_F/Z_F. While T and T_F have some structural and operational similarities, they behave fundamentally differently, particularly with regards to dynamics. In this dissertation we seek analogues to known results in R and T. We find that both sets have similar ergodicity results but that orbits of affine maps and other areas of dynamics show significant differences. In particular, we construct an Artin-Mazur zeta function that looks significantly different to its counterpart in T, demonstrate that Furstenberg's orbital density theorem falls apart in positive characteristic, and establish that the intersection of orbits of affine maps rely on sets that depend on powers of the characteristic of F rather than arithmetic progressions. At first glance, the simplicity of working in T_F and its similarities to T suggest that we should be able to find many of the same simple results; however in reality the structure of T_F consists of infinitely defined sub-structures constructed by shifts of Frobenius maps into itself and these sub-structures present themselves frequently in a manner that does not occur in T
dc.identifier.citationGunn, K. (2024). Some problems on the dynamics of positive characteristic tori (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.
dc.identifier.urihttps://hdl.handle.net/1880/118760
dc.identifier.urihttps://doi.org/10.11575/PRISM/46357
dc.language.isoen
dc.publisher.facultyGraduate Studies
dc.publisher.institutionUniversity of Calgary
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.
dc.subjectNumber Theory
dc.subjectTori
dc.subjectDynamics
dc.subjectPositive Characteristic
dc.subject.classificationMathematics
dc.titleSome Problems on the Dynamics of Positive Characteristic Tori
dc.typedoctoral thesis
thesis.degree.disciplineMathematics & Statistics
thesis.degree.grantorUniversity of Calgary
thesis.degree.nameDoctor of Philosophy (PhD)
ucalgary.thesis.accesssetbystudentI do not require a thesis withhold – my thesis will have open access and can be viewed and downloaded publicly as soon as possible.
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
ucalgary_2024_gunn_keira.pdf
Size:
708.94 KB
Format:
Adobe Portable Document Format
Download
License bundle
Now showing 1 - 1 of 1
No Thumbnail Available
Name:
license.txt
Size:
2.62 KB
Format:
Item-specific license agreed upon to submission
Description:
Download
Collections
Open Theses and Dissertations